HMMT 二月 2012 · 代数 · 第 3 题
HMMT February 2012 — Algebra — Problem 3
题目详情
英文原题
Given points aand bin the plane, let a⊕bbe the unique point csuch that abcis an equilateral triangle with a, b, c in the clockwise orientation. Solve ( x⊕(0,0))⊕(1,1) = (1 ,−1) for x.
解析
英文解析
- Given points a and b in the plane, let a ⊕ b be the unique point c such that abc is an equilateral trianglewith a, b, c in the clockwise orientation.
Solve ( x ⊕ (0 , 0)) ⊕ (1 , 1) = (1 , − 1) for x .
√ √
1 − 3 3 − 3
Answer: ( , ) It is clear from the definition of ⊕ that b ⊕ ( a ⊕ b ) = a and if a ⊕ b = c
2 2
√
then b ⊕ c = a and c ⊕ a = b . Therefore x ⊕ (0 , 0) = (1 , 1) ⊕ (1 , − 1) = (1 − 3 , 0). Now this means
√ √
√
1 − 3 3 − 3
x = (0 , 0) ⊕ (1 − 3 , 0) = ( , ).
2 2